The Canonical Exact Sequence of Differential Modules for 0-Dimensional Schemes

Given a 0-dimensional scheme $\text{X}$ in ${\mathbb{P}}_K^n$ over a perfect field $K$, we examine the second differential power of its homogeneous vanishing ideal. This enables us to establish the canonical exact sequence for the associated Kähler differential module. We also provide a formula for the Hilbert polynomial of Kähler differential modules when $\text{X}$ is either a fat point scheme or a 0-dimensional locally monomial Gorenstein scheme.